Continuous Wavelet Transforms
The WFT localizes a signal simultaneously in time and frequency by "looking" at it through a window that is translated in time, then translated in frequency (i.e., modulated in time). These two operations give rise to the "notes" gω,t(u). The signal is then reconstructed as a superposition of such notes, with the WFT ƒ(tω,t) as the coefficient function. Consequently, any features of the signal involving time intervals much shorter than the width T of the window are underlocalized in time and must be obtained as a result of constructive and destructive interference between the notes, which means that "many notes" must be used and ƒ(ω, t) must be spread out in frequency. Similarly, any features of the signal involving time intervals much longer than T are overlocalized in time, and their construction must again use "many notes," with ƒ(ω,t) spread out in time. This can make the WFT an inefficient tool for analyzing regular time behavior that is either very rapid or very slow relative to T. The wavelet transform solves both of these problems by replacing modulation with scaling to achieve frequency localization.
KeywordsInverse Fourier Transform Mother Wavelet Continuous Wavelet Transform Continuous Wavelet Coefficient Function
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